Method for shared estimation of parameters

ABSTRACT

A method for shared estimation of parameters (ε, φ, |C 1 |, C 0 , α, Δω) is described, which together with an error vector e(k), describe the connection between a digitally modulated reference signal inputted to a transmission channel and a received receiver signal z(k) which is at an end of the transmission channel. The method includes the following steps: forming the error vector e(k) in dependence of the parameters (ε, φ, |C 1 |, C 0 , α, Δω), a reference signal s(k), and the receiver signal z(k); linearizing the error vector e(k); substituting a real parameter of the linearized error vector through components of a estimation vector, wherein a substituted error vector is produced; inserting the substituted error vector into the cost function; and determining the estimation vector through gradient development of the cost function and subsequently setting the gradient to zero.

[0001] This nonprovisional application claims priority under 35 U.S.C. §119(a) on Patent Application No. 101 57 247.6 filed in Germany on Nov. 22, 2001, which is herein incorporated by reference.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates to a method and computer program for shared estimation of parameters, in particular to the determination of “Error Vector Magnitude” (hereinafter “EVM”).

[0004] 2. Description of the Background Art

[0005] The “Error Vector Magnitude” (EVM) is used often, to estimate the linearity of a digitally modulated mobile radio system. For Example, the standard “GSM 05.05, version 8.5.0, Draft ETSI EN 300 910 V.8.5.0, (2000-07), Annex G” (hereinafter “the standard”) defines the requirements on the EVM for the 8-PSK GSM EDGE-System. However, the standard does not define algorithms in order to determine the EVM.

[0006]FIG. 1 shows an example configuration, in accordance with the above given standard, of a transmission channel 20 having different parameters ε, w, C₁, and C₀. The parameter ε represents a time shift, that determines the signal in the delay 21 of the configuration, e(k) is an error vector which is added in the configuration in a adder 22, C₁ is a complex amplification which is added in the multiplier 23, and C₀ represents a constant level of a DC-offset. The parameter W^(k) is used to model the time response during a burst (transmit block), for example due to a heating up of the amplifier.

[0007] The following allocations are applied:

[0008] T_(s): symbol period;

[0009]  The sequence of the configuration, are present before the symbol clock of the time points kT_(s).

[0010] s(k): reference signal: is a trouble-free input signal after the measurement filter in the receiver to the symbol-time points kT_(s);

[0011] e(k): error vector;

[0012] ε: resultant time shift due to a non-ideal estimation of the preceding coarse time shift estimation;

[0013] C₁: complex amplification (gain) of the measurement signal;

[0014] C₀: constant level of a (DC) offset in the measurement signal; and

[0015] w=e^(α+jΔωT) ^(_(s)) α describes the change in amplitude of the measurement signal, which, for example, because of the heating up of the amplifier, causes a higher signal level to arise within the bursts.

[0016]  Furthermore, through Δω the resultant frequency shift is modeled due to the preceding non-ideal coarse frequency estimation.

[0017] From the reference signal s(k), the following received signal z(k) results:

z(k)={C ₀ +C ₁ ·[s(k)+e(k)]}·w ^(k).  (1)

[0018] The error vector e(k) is determined by: $\begin{matrix} {{e(k)} = {\frac{{z(k)} \cdot w^{- k}}{C_{1}} - \frac{C_{0}}{C_{1}} - {{s\left( {k - ɛ} \right)}.}}} & (2) \end{matrix}$

[0019] In view of this model, a total of seven real parameters have to be estimated. It is noted that the time shift ε, in view of the excessive sampled sequence, is to be understood to fulfill the sample theorem.

[0020] The “Error-Vector magnitude” (EVM) is calculated over a burst and is defined as follows: $\begin{matrix} {{EVM} = {\sqrt{\sum\limits_{k}\left| {e(k)} \middle| {}_{2}{/\sum\limits_{k}} \middle| {s(k)} \right|^{2}}.}} & (3) \end{matrix}$

[0021] To determine the “Error-Vector magnitude” (EVM), the parameters ε, C₀, C₁ and w, must be estimated such that a minimum of the “Error-Vector magnitude” (EVM) per burst results. Through the use of this parameter, the individual error vector e(k) can be calculated for each symbol.

[0022] The article “A Method for Computing Error Vector Magnitude in GSM EDGE Systems—Simulation Results,” IEEE Communications Letters, VOL. 5, NO. 3, March 2001, pages 88 to 91, discloses a method to determine the parameters ε, C₀, C₁ and w. This conventional method is, however, not very efficient. For one, the conventional method has the disadvantage that the time shift ε is not subjected to a common estimation with the parameters C₀, C₁, and w, but that only a coarse estimation of the time shift ε is performed before the common estimation of the parameters C₀, C₁ and w. Moreover, the conventional method has the disadvantage that a gradient method has to be used, which converges relatively slowly. The conventional method requires therefore, a relatively large number of iterations, which are dependent on the arbitrary start values for C₀, C₁ and w.

SUMMARY OF THE INVENTION

[0023] It is, therefore, an object of the present invention to provide a method to commonly estimate several parameters, which together with the error vector describes the connection between a digitally modulated reference signal fed through the transmission channel and a received signal that is received at the end of the transmission channel, which method utilizes a fewer number of iterations and converges quickly. The invention also provides a computer program to execute the method.

[0024] A preferred embodiment of the method of the invention, uses a linearization and a substitution to calculate the parameters analytically. Because of linearization, a minor error arises. Through iterative repetition, the linearization error can be desirably reduced. In the norm, two iterations are sufficient. These are substantially fewer iterations than those needed in the conventional method.

[0025] Further scope of applicability of the present invention will become apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The present invention will become more fully understood from the detailed description given herein below and the accompanying drawings which are given by way of illustration only, and thus, are not limitive of the present invention, and wherein:

[0027]FIG. 1 is a configuration of a transmission channel;

[0028]FIG. 2 is a table which represents the elements M_(i,j) of the Matrix M and the components b_(i) of the vector b; and

[0029]FIG. 3 is a block diagram, illustrating a preferred embodiment.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0030] The estimation method according to a preferred embodiment of the invention is described herein below, wherein estimated parameters are a time shift ε, a phase shift φ, a amplification |C₁|, a constant level shift C₀, a amplitude change α, and a frequency shift Δω, which determine a reference signal s(k) in the transmission channel. The method according to the invention is not limited to this application example, and is also suitable for the estimation of other parameters, which characterize the transmission channel.

[0031] The estimated parameters are determined through the minimization of the cost function: $\begin{matrix} {{L\left( \overset{\sim}{x} \right)} = {\sum\limits_{k}\left| {\overset{\sim}{e}(k)} \right|^{2}}} & (4) \end{matrix}$

[0032] wherein K is the symbol number within the evaluation area (“useful part,” e.g. a burst). Generally, in this invention, a test parameter is represented by a “snake” and the estimated parameter is represented by a “roof,” that is, {circumflex over (x)} describes generally the parameter that is to be estimated, and {tilde over (e)}(k) describes a therefrom resultant test-error vector.

[0033] Subsequently, the estimation according to a preferred embodiment is deduced. In the deduction, for the purpose of improving the overview, no iteration specific nomenclature is used.

[0034] In C₁ the amplification (gain) |C₁| and the remaining phase shift φ is modeled through the non-ideal preceding phase compensation according to:

C ₁ =|C ₁ |·e ^(jφ)  (5)

[0035] In w, according to:

w=e ^(α+jΔωT) ^(_(s)) .  (6)

[0036] the amplitude change α in the measurement signal and the resultant frequency shift Δω, is modeled.

[0037] By substituting the equations (5) and (6) into equation (2) the error vector is determined $\begin{matrix} {{e(k)} = {{\frac{1}{\left| C_{1} \right|} \cdot {z(k)} \cdot e^{{{- \alpha}\quad k} - {j\quad \Delta \quad \omega \quad {kT}_{s - {j\quad \phi}}}}} - \frac{C_{0}}{C_{1}} - {s\left( {k - ɛ} \right)}}} & (7) \end{matrix}$

[0038] Due to the preceding coarse estimation, a linearization in equation (7) is allowable: for a complex x, via Taylor series expansion, e^(x) is generally linearized by

e ^(x)≈1+x.  (8)

[0039] Furthermore, the time shift s(k−ε) is linearized by

s(k−ε)=s(k)−ε·s _(d)(k).  (9)

[0040] It is noted that the normed derivative s_(d)(k) is not allowed to be calculated from s(k), because s(k) does not fulfill the sample theorem. Rather, the over-sampled sequence s_(ov)(k) should be used.

[0041] By substituting equations (8) and (9) into equation (7), the linearization error vector is obtained: $\begin{matrix} {{e(k)} = {{{\frac{1}{\left| C_{1} \right|} \cdot z}{(k) \cdot \left\lbrack {1 - {\alpha \quad k} - {{j \cdot \Delta}\quad \omega \quad {kTs}} - {j\quad \phi}} \right\rbrack}} - \frac{C_{0}}{C_{1}} - {\left\lbrack {{s(k)} - {ɛ \cdot {s_{d}(k)}}} \right\rbrack.}}} & (10) \end{matrix}$

[0042] Through substitution, the real parameters x_(i) of the vector x is determined, according to: $\begin{matrix} {{e(k)} = {{{z(k)} \cdot \left\lbrack {\frac{1}{\underset{\underset{x_{1}}{}}{\left| C_{1} \right|}} - {\frac{\alpha}{\underset{\underset{x_{7}}{}}{\left| C_{1} \right|}} \cdot k} - {j \cdot \frac{\Delta \quad \omega \quad T_{s}}{\underset{\underset{x_{2}}{}}{\left| C_{1} \right|}} \cdot k} - {j \cdot \frac{\phi}{\underset{\underset{x_{3}}{}}{\left| C_{1} \right|}}}} \right\rbrack} - \frac{C_{0}}{\underset{\underset{x_{4} + {j \cdot x_{5}}}{}}{C_{1}}} + {\underset{\underset{x_{6}}{}}{ɛ} \cdot {s_{d}(k)}} - {{s(k)}.}}} & (11) \end{matrix}$

[0043] The real estimation value vector is defined through:

x=(x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇)^(T)

[0044] Through conversion, according to equation (11), the estimation values to be focused on from the estimation value vector x are determined according to:

|C ₁|=1/x ₁

ΔωT _(s) =x ₂ ·|C ₁|

Δφ=x ₃ ·|C ₁|

C ₁ =|C ₁ |·e ^(jφ)

C ₀ =C ₁·(x ₄ +j·x ₅)

ε=x₆

α=x ₇ ·|C ₁|  (12)

[0045] by defining the function f_(i)(k), equation (10) results in: ${e(k)} = {{\sum\limits_{i = 1}^{N}\quad {x_{i} \cdot {f_{i}(k)}}} - {s(k)}}$

[0046] with

f ₁(k)=z(k) f ₂(k)=−j·k·z(k) f ₃(k)=−j·z(k) f ₄(k)=−1 f ₅(k)=−j f ₆(k)=s _(d)(k) f ₇(k)=−z(k)·k  (13)

[0047] Through gradient development, the cost function L(x) and the subsequent zero setting of the gradient, the estimation value vector {circumflex over (x)} is obtained according to:

{circumflex over (x)}=M ⁻¹ ·b  (14)

[0048] with the matrix and vector elements being: $\begin{matrix} \begin{matrix} {M_{ij} = {{Re}\left\{ {\sum\limits_{k}{{f_{i}^{*}(k)} \cdot {f_{j}(k)}}} \right\}}} \\ {b_{i} = {{Re}{\left\{ {\sum\limits_{k}{{f_{i}^{*}(k)} \cdot {s(k)}}} \right\}.}}} \end{matrix} & (15) \end{matrix}$

[0049] By substituting equation (13) into equation (15) the Matrix M and the Vector b is obtained, which is shown in FIG. 2.

[0050] By linearizing equation (8) and equation (9), the error-prone to the estimation vector {circumflex over (x)} in equation (14) is negligible. According to a preferred advancement of the method of the invention, several iterations are performed. The error can be desirably reduced through several iterations. In the norm, the error is negligible after 2 iterations.

[0051] The following is applied: Iteratio: The number of iterations to be performed; loop=[1,Iteration]: The parameter loop shows which iteration is currently being performed; x^((loop)): The index ^((loop)) describes the value x of the loop-ed iteration (example: {circumflex over (ε)}^((loop)), M^((loop)); z_(ov) ^((loop))(K): Over-sampled measurement signal of the loop-ed iteration, with the linearized estimated parameters compensated; and z^((comp))(k): Measurement signal, with all estimated parameters compensated. From this sequence, the EVM-error E{circumflex over (V)}M(k) is calculated.

[0052]FIG. 3 is a block diagram illustrating the iterative method for estimating parameters. Before the embodied refined parameter estimation, a coarse estimation and compensation of the frequency ω, phase φ, and time shift ε, has to be performed.

[0053] On the inputs 2 and 3 of a refined estimator 1, the over-sampled measurement sequence (receiver sequence) z_(ov)(k), and the reference sequence s_(ov)(k), are given, respectively. On outputs 4 a and 4 b the estimated parameters are present, and on an output 5 the relative C₀ and C₁ from a compensator 18 having a compensator receiver sequence z^((comp))(k) (in a symbol pulse), are present.

[0054] From the over-sampled reference signal s_(ov)(k), the normed differential sequence is calculated with the impulse answer h_(diff)(k) in a filter 6. Subsequently, in a reducing rate sampler 7 the down-sampling-factor ov is down sampled, followed by time slotting by a multiplier 8. Therewith, on the input 9 of the estimation block 10, lies the sequence s_(d)(k) in a symbol pulse. Through a sampling rate reducer 11 and a multiplier 12, in which the time slotting takes place, the unfiltered, down-sampled and slotted reference sequence s(k) can be fed to an input 17 of the estimation block 10.

[0055] For the estimation, only the valid symbols (“useful symbols”) are used, wherefore a slotting before the estimation must be performed. In the over-sampled input signals a pre-run and post-run are needed. The reason therefore is that a FIR (finite impulse response) filter 6 needs a rise time for the differentiation and also for the non-pictured interpolation filter, to compensate the estimated time shift {circumflex over (ε)}. If a plurality of iterations are performed, the measurement sequence z_(ov)(k), at the beginning of the next iteration, has to be compensated with the actual total estimation value in a compensator 13, and before the compensated measurement sequence z_(ov) ^((loop))(k) is fed to an input 16, it goes through the reducing rate sampler 14 and the multiplier 15, in which the time slotting takes place.

[0056] The following points have to be thereby regarded:

[0057] Only the linearized estimation parameters ({circumflex over (ε)}, ŵ, and {circumflex over (φ)}) are compensated at the beginning of a new iteration.

[0058] For the linearized estimation parameters it applies that: the linearized (refined)-estimation value of the loop-ed iteration are {circumflex over (ε)}^((loop)), ŵ^((loop)) and {circumflex over (φ)}^((loop)). The total estimation value according to the loop-ed iteration results through addition of all of the past estimation values according to: $\begin{matrix} \begin{matrix} {\hat{ɛ} = {\sum\limits_{l = 1}^{loop}\quad {\hat{ɛ}}^{(l)}}} \\ {\hat{w} = {\sum\limits_{l = 1}^{loop}\quad {\hat{w}}^{(l)}}} \\ {\hat{\phi} = {\sum\limits_{l = 1}^{loop}\quad {\hat{\phi}}^{(l)}}} \end{matrix} & (16) \end{matrix}$

[0059] With these instantaneous total estimation values, the measurement sequence will be compensated in the next iteration.

[0060] With every new iteration, the measurement sequence z_(ov)(k) is compensated with the actual total estimation values {circumflex over (ε)}, ŵ and {circumflex over (φ)} of the linearized parameters.

[0061] The non-linearized estimation values (Ĉ₀ and |Ĉ₁|) are not compensated for, in the single iterations, but are newly calculated in every iteration. Otherwise, false reproductions may arise because of the linearization error in the single iterations.

[0062] It is to be heeded, that the estimated time shift {circumflex over (ε)} is not compensated in the reference signal, but in the measurement signal (input signal). Through which, according to the measurement regulation, the standards are achieved, in that the measurement signal is interpolated on the inter-symbol-interference-free symbol time point.

[0063] After the last iteration loop-iteration, the equation (16) of the resultant total estimation value of the linearized parameter is present. The non-linearized parameters are taken from the calculation of the last iteration. Finally, the compensated measurement sequence z^((comp))(k) in FIG. 3, must be calculated, which is needed for the calculation of the estimated EVM-vector, according to: $\begin{matrix} {{z^{({comp})}(k)} = {\frac{z^{({loop})}(k)}{\left| {\hat{C}}_{1} \right|} - \frac{{\hat{C}}_{0}}{{\hat{C}}_{1}}}} & (17) \end{matrix}$

[0064] When the method according to the invention is utilized in a CDMA (Code Division Multiple Access) signal, the reference signal comprises a plurality of superimposed partial signals from different code channels and always one parameter for every partial signal describes the different amplifications of the different code channels. The amplification-parameter of the different partial signals is estimated simultaneously with the method according to the invention.

[0065] The invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and all such modifications as would be obvious to one skilled in the art are to be included within the scope of the following claims. 

What is claimed is:
 1. A method for shared estimation of a plurality of parameters (ε, φ, |C₁|, C₀, α, Δω), which together with an error vector e(k), describe the connection between a digitally modulated reference signal inputted to a transmission channel and a received receiver signal z(k) which is at an end of the transmission channel, said method comprising the steps of: forming the error vector e(k) in dependence of the parameters (ε, φ, |C₁|, C₀, α, Δω), a reference signal s(k), and the receiver signal z(k); linearizing the error vector e(k); substituting a real parameter of the linearized error vector through components of an estimation vector x=(x₁x₂x₃x₄x₅x₆x₇), wherein a substituted error vector in the form of ${e(k)} = {{\sum\limits_{i = 1}^{N}{x_{i} \cdot {f_{i}(k)}}} - {s(k)}}$

 is produced; inserting the substituted error vector into the cost function ${{L\left( \overset{\sim}{x} \right)} = {\sum\limits_{K}\left| {\overset{\sim}{e}(k)} \right|^{2}}};$

 and determining the estimation vector {circumflex over (x)} through gradient development of the cost function L(x) and subsequently setting the gradient to zero.
 2. The method according to claim 1, wherein the method steps are at least once iteratively repeated, to reduce the linearization error.
 3. The method according to claim 1, wherein the parameters (ε, φ, |C₁|, C₀, α, Δω) are a time shift ε, a phase shift φ, a amplification |C₁|, a constant level shift C₀, a amplitude change α, and a frequency shift Δω, which determine the reference signal s(k) on the transmission channel.
 4. The method according to claim 3, wherein the error vector e(k), in dependence of the parameters (ε, φ, |C₁|, C₀, α, Δω), the reference signal s(k), and the received signal z(k), has the form ${{e(k)} = {{\frac{1}{\left| C_{1} \right|} \cdot {z(k)} \cdot e^{{{- \alpha}\quad k} - {j\quad \Delta \quad \omega \quad {kT}_{s}} - {j\quad \phi}}} - \frac{C_{0}}{C_{1}} - {s\left( {k - ɛ} \right)}}},$

wherein T_(s) is the symbol period, and a signal sequence, is in a signal pulse, and is present to time points k·T, and wherein C₁=|C₁|·e^(jφ) is applied.
 5. The method according to claim 4, wherein through linearization, an exponential function of the error vector e(k), by Taylor series expansion, is linearized to the first order, such that the linearized error vector results in ${e(k)} = {{\frac{1}{\left| C_{1} \right|} \cdot {z(k)} \cdot \left\lbrack {1 - {\alpha \quad k} - {{j \cdot {\Delta\omega}}\quad {kT}_{s}} - {j\quad \phi}} \right\rbrack} - \frac{C_{0}}{C_{1}} - {\left\lbrack {{s(k)} - {ɛ \cdot {s_{d}(k)}}} \right\rbrack.}}$


6. The method according to claim 5, wherein in the linearized error vector the substitutions |C ₁|=1/x ₁ ΔωT _(s) =x ₂ ·|C ₁|Δφ=x ₃ ·|C ₁|C ₁ =|C ₁ |·e ^(jφ) C ₀ =C ₁·(x ₄ +j·x ₅) ε=x₆ α=x ₇ ·|C ₁|are made, to produce the estimation vector: x=(x ₁ x ₂ x ₃ x ₄ x ₅ x ₆ x ₇)^(T).
 7. The method according to claim 1, wherein the reference signal is a GSM EDGE-Signal, in particular a mobile radio signal, that is defined in GSM 05.05, version 8.5.0, Draft ETSI EN 300 910 V.8.5.0, (2000-07), Annex G.
 8. The method according to claim 1, wherein the reference signal comprises a plurality of superimposed partial signals from different code channels and wherein always at least one parameter for every partial signal describes the different amplifications of the different code channels.
 9. The method according to claim 8, wherein the reference signal is a CDMA signal, in particular a mobile radio signal that is defined by the standards 3GPP TS 25.141, 3GPP2 C.S0010-A, 3GPP2 C.S0011-A, 3GPP2 C.P9011, or 3GPP2 C.P9012.
 10. A computer program with program code resources to perform all of the steps according to claim 1, when the computer program is executed on a computer or a digital signal processor.
 11. A computer program with program code resources to perform all of the steps according to claim 1, when the computer program is stored on a machine-readable data medium.
 12. A computer program product, having program code resources stored on a machine readable medium, to perform all of the steps according to claim 1, when the program is executed on a computer or a digital signal processor. 